well for starters, the total thermal noise in the noise bandwidth of your circuit, or in case of a sampled circuit, the total thermal noise present in a sample will be a function of the form

alpha*kT/Ceffective, where alpha is a constant dependent on your circuit parameters and Ceffective the effective capacitance that determines the noise.

So the total thermal noise power is actually always (1/2)*kT (if you have optimized your opamp and feedback to be as good as ideally possible). Using a larger cap simply reduces the voltage noise levels (V^2), but not the noise power. What you do now is to supply more signal power if you are trying to achieve a certain signal swing (say 1 V), since that 1V swing is now being realized over a larger cap. If you are trying to achieve the same fixed bandwidth, you are also raising the gm now.

Increasing gm on the other hand allows you to reduce the noise spectral density, i.e. lower noise level, but higher noise bandwidth if you don't change the load cap, i.e. same total noise power. In a continuous-time circuit where the noise bandwidth is not determined by the noise-critical stage, e.g. you have a narrow filter somewhere down the chain, the reduction in noise spectral density translates to net total noise power.

To rephrase:

Total noise power = alpha*kT/Ceff
More Ceff => lower noise power
More Ceff, fixed bandwidth => more power/gm needed in whatever is driving the Ceff => SNR is raised because Signal power is increased
More gm => lower noise spectral density, but not necessarily lower total noise power.

What you need to watch out for, especially in switched cap circuits:

Ceffective is usually the capacitance that limits the bandwidth of your circuit, i.e. not just the load cap. By the way, you need to be careful how you define the load cap in case you are talking of a single-stage OTA. The load also consists of the feedback network for instance.

alpha will typically depend on the noise factor of your opamp, how large the gm of your input devices is compared to the gm of the current mirrors etc. for instance, and also on the switch resistance etc.

The total noise power in a switched circuit is very dependent on the way you do the switching and so you must be very careful in trying to calculate/simulate it. If you do it well, you will also see automatically whether tweaking the operation of one or the other phase allows possibility for sinking the total noise level.

Vivek